The hypersurface
in
the
sphere
Yang
Jiang
School
of
Mathematics
and
Statistics, Northeast Normal University
Abstract
We consider hypersurfaces in the unit lightlike sphere. The unit sphere can be
canonically embedded in the lightcone and de Sitter space in Minkowski space. We
investigate thesehypersurfaces intheframeworkofthe theory of Legendriandualities
betweenpseudo-spheresinMinkowski space. This isan anouncement of the results in [15]
1.
Introduction
In [2, 3], professor Izumiya has introduced the mandala of Legendrian dualities between pseudo-spheres in Minkowski space. There are three kinds of pseud$(\succ$spheres in Minkowski
space (i.e.,Hyperbolicspace,deSitter space and thelightcone). Especially, if
we
investigatespacelike submanifolds in the lightcone, those Legendrian dualities are essentially useful (see, also [7]). Forde Sitter spaceand thelightcone in Minkowski $(n+2)$
-space,
there existnaturally embedded unit $r\succ$-spheres. Moreover,
we
have the canonical projection from thelightcone to the unit sphereembedded in the lightcone(cf.,
\S 2).
In thispaper weinvestigate hypersurfaces in the unit $n$-sphere in the framework of the theory of Legendrian dualities between pseudo-spheres in Minkowski $(n+2)$-space ([3, 4, 12, 13], etc.). If we have ahypersurface in the unit $\gamma\triangleright$sphere, then we have spacelike hypersurfaces in the embedded
unit $n$-sphere in the lightcone and de Sitter space. Therefore, we naturally have the dual hypersurfaces in the lightcone as an application of the duality theorem in [3]. There are
two kinds of lightcone dual hypersurfaces ofahypersurface in the unit $n$-sphere. One is the dual ofthe hypersurface ofthe unit $n$-sphere embedded in de Sitter space and another is
the dualofthe hypersurface of theunit $n$-sphereembedded in the lightcone. By definition, these dual hypersurfaces are different.
On the other hand,
we
havestudied
thecurves
in the unit 2-sphere and the unit 3-spherefrom the view point of the Legendrian duality in [5, 6]. In the unit 2-sphere, it is known
that the evolute of
a curve
in the unit 2-sphere is the dual of the tangent indicatrix of theoriginal curve [11]. We have shown that the projection images of the critical value sets of lightcone dual surfaces for a curve in the unit 2-sphere coincide with the evolute of the original curve in [5]. However, this fact doesn’t hold for a curve in unit 3-sphere (cf., [6]). For the curve case, these facts has been shown by the direct calculations in [5, 6]. We have not known the geometric
reason
why the situations are different. In order to clarify these situation, weinvestigatehypersurfaces in the unit $n$-sphere from the viewpoint ofthetheory ofLegendrian singularities. Thecurves in theunit 2-sphere can be considered as a
special
case
ofthis paper. Wecan
also show that the projection images of the critical value sets of two different lightcone dual hypersurfaces for a hypersurface in the unit $n$-sphere also coincide with the spherical evolute (cf., [10]) of the original hypersurface. We interpretgeometric meanings of the singularities of those two lightcone dual hypersurfaces. Here, we remark that we do not have the notion of tangent indicatrices for higher dimensional submanifolds in the sphere. Therefore, the situation is completely different from the curve
case. In [15], we givea classification of thegeneric singularities ofthe lightcone duals of the
surface in the unit 3-sphere.
All maps and submanifolds considered here areofclass $C^{\infty}$ unless otherwise stated.
2.
The
basic
concepts
Let$\mathbb{R}^{n+2}$bean$(n+2)$-dimensional vector space. Forany
two vectors$x=(x_{0}, x_{1}, \ldots, x_{n+1}),$ $y=$ $(y_{0}, y_{1}, \ldots, y_{n+1})$ in $\mathbb{R}^{n+2}$, their pseudo scalar
product is defined by $\langle x,$$y\rangle=-x_{0}y_{0}+$
$x_{1}y_{1}+\ldots+x_{n+1}y_{n+1}$. Here, $(\mathbb{R}^{n+2}, \langle)\rangle)$ is called Lorentz-Minkowski $(n+2)$-space (simply,
Minkowski$(n+2)$-space),which is denoted by$\mathbb{R}_{1}^{n+2}$.Forany $(n+1)$vectors
$x_{1},$ $x_{2},$$\ldots,$$x_{n+1}\in$
$\mathbb{R}_{1}^{n+2}$, their pseudo vector product is defined by
$x_{1}\wedge x_{2}\wedge\ldots\wedge x_{n+1}=$
$-e_{0}$ $e_{1}$ . . . $e_{n+1}$ $x_{1}^{0}$ $x_{1}^{1}$ . .
.
$x_{1}^{n+1}$$x_{2}^{0}$ $x_{2}^{1}$
.
.
.
$x_{2}^{n+1}$. . .
$x_{n+1}^{0}$ $x_{n+1}^{1}$ . .
.
$x_{n+1}^{n+1}$where $\{e_{0}, e_{1}, \cdots, e_{n+1}\}$ is the canonical basis of $\mathbb{R}_{1}^{n+2}$ and $x_{i}=(x_{i}^{0},x_{i}^{1}, \cdots,x_{i}^{n+1})$. $A$
non-zerovector $x\in \mathbb{R}_{1}^{n+2}$ is calledspacelike, lightlike or timelike if $\langle x,$$x\rangle>0,$$\langle x,$$x\rangle=0$ or
$\langle x,$$x\rangle<0$ respectively. The norm of$x\in \mathbb{R}_{1}^{n+2}$ is defined by $\Vert x\Vert=\sqrt{|\langle x,x\rangle|}$
.
We define the de Sitter $(n+1)$-space by$S_{1}^{n+1}=\{x\in \mathbb{R}_{1}^{n+2}|\langle x, x\rangle=1\}.$
We define the dosed lightcone with the vertex$a$ by
$LC_{a}=\{x\in \mathbb{R}_{1}^{n+2}|\langle x-a,x-a\rangle=0\}.$
We define the open lightcone at the onginby
$LC^{*}=\{x\in \mathbb{R}_{1}^{n+2}\backslash \{0\}|\langle x, x\rangle=0\}.$
We consider a submanifold in the lightcone defined by $S_{+}^{n}=\{x\in LC^{*}|x_{1}=1\}$, which is
called the lightlike unit sphere. We have a projection $\pi$ : $LC^{*}arrow S_{+}^{n}$ defined by
$\pi(x)=\tilde{x}=(1,\frac{x_{1}}{x_{0}}, \ldots, \frac{x_{n+1}}{x_{0}})$ ,
where$x=(x_{0}, x_{1}, \ldots x_{n+1})$.We also define the$n$-dimensional Euclideanunitsphere in$\mathbb{R}_{0}^{n+1}$
by $S_{0}^{n}=\{x\in S_{1}^{n+1}|x_{0}=0\}$, where $\mathbb{R}_{0}^{n+1}=\{x\in \mathbb{R}_{1}^{n+2}|x_{0}=0\}.$
Let $x$ : $Uarrow S_{+}^{n}$ be an embedding from an open set $U\subset \mathbb{R}^{n-1}$
.
We identify $M=x(U)$with $U$ through the embedding $x$
.
Obviously, the tangent space$T_{p}M$ are all spacelike (i.e.,
consists only spacelike vectors), so $M$ is a spacelike hypersuface in $S_{+}^{n}\subset \mathbb{R}_{1}^{n+2}$
.
We havea map $\Phi$ :
$S_{+}^{n}arrow S_{0}^{n}$ defined by $\Phi(v)=v-e_{0}$, which is an isometry. Then we have a
hypersurface$\overline{x}$: $Uarrow S_{0}^{n}$ defined by
same
geometric propertiesas
spherical hypersurfaces. For any $p=x(u)$, wecan
construct a unit normal vector$n(u)$as
$n(u)= \frac{\overline{x}(u)\wedge e_{0}\wedge x_{u_{1}}(u)\wedge..\cdot\wedge x_{u_{n-1}}(u)}{\Vert\overline{x}(u)\wedge e_{0}\wedge x_{u_{1}}(u)\wedge..\wedge x_{u_{n-1}}(u)\Vert}.$
We have $\langle n(u),$$n(u)\rangle=1,$ $\langle e_{0},$$e_{0}\rangle=-1$ and $\langle e_{0},$$n\rangle=\langle n,x_{u_{i}}\rangle=\langle n,x\rangle=0$
.
The system$\{e_{0},n(u),\overline{x}(u),x_{u_{1}}(u), \ldots, x_{u_{n-1}}(u)\}$is a basisof $l_{p}’\mathbb{R}_{1}^{n+2}$
.
We definea
map$G:Uarrow S_{0}^{n}$by$G(u)=n(u)$. We call it the Gauss mapofthe hypersurface$M=x(U)$. We have alinear mapping provided by the derivation of the Gauss map at $p\in M,$ $dG(u)$ : $T_{p}Marrow T_{p}M.$
We call the linear transformation$S_{p}=-dG(u)$ the shape operatorof $M$ at $p=x(u)$
.
Theeigenvalues of$S_{p}$ denoted by $\{\kappa_{i}(p)\}_{i=1}^{n-1}$ are called the principal curvatures of$M$ at $p$
.
TheGauss-Kronecker curvatureof$M$ at$p$is defined to be$K(p)=\det S_{p}.$ $A$ point$p$is called
an
umbdic point ifall theprincipal curvatures coincide at $p$ andthus wehave $S_{p}=\kappa(p)id_{1_{p}M}$
forsome $\kappa(p)\in \mathbb{R}$
.
We saythat$M$is totallyumbilic ifallthe points on$M$areumbilic. Since$x$ is aspacelike embedding, we have a$Ri$emannian metric (or the
first
fundamenta)form)on
$M$ given by $ds^{2}= \sum_{i,j=1}^{n-1}g_{ij}du_{i}du_{j}$, where $g_{ij}(u)=\langle x_{u_{i}}(u),$$x_{u_{j}}(u)\rangle$ for any $u\in U.$The secondfundamental form on $M$ is given by$h_{ij}(u)=-\langle n_{u_{t}}(u),$$x_{u_{j}}(u)\rangle$ at any $u\in U.$
Under the above notation, we have the following Weingarten formula [9]:
$G_{u_{t}}=- \sum_{j=1}^{n-1}h_{i}^{j}x_{u_{j}}(i=1, \ldots, n-1)$,
where $(h_{i}^{j})=(h_{ik})(g^{kj})$ and $(g^{kj})=(g_{kj})^{-1}$. This formula induces an explicit expression
of the Gauss-Kroneckercurvature in terms of the Riemannianmetricand the second
funda-mentalinvariant given by $K=det(h_{ij}/det(g_{\alpha\beta})$
.
$A$ point $p$isa
parabolic point if$K(p)=0.$Apoint $p$is a
flat
pointif it isan
umbilicpoint and $K(p)=0.$In [10] the spherical evolute of a hypersurface has been introduced and investigated the singularities. Each spherical evolute of$\overline{M}=\overline{x}(U)$ isdefined to be
$\epsilon\frac{\pm}{M}=\bigcup_{i=1}^{n-1}\{\pm(\sqrt{\frac{\kappa_{t}^{2}(p)}{1+\kappa_{i}^{2}(p)}}\overline{x}(u)+\sqrt{\frac{1}{1+\kappa_{i}^{2}(p)}}n(u))|p=x(u)\in M=x(U)\}.$
3.
The lightcone dual surfaces and the
lightcone height
functions
In [3], professor Izumiya hasintroducedtheLegendrian dualitiesbetween pseudo-spheres
in Minkowski space which is a basic tool for the study of hypersurfaces in pseudo-spheres
in Minkowski space. We define one-forms $\langle dv,$$w \rangle=-w_{0}dv_{0}+\sum_{i=1}^{n+1}w_{i}dv_{i},$ $\langle v,$$dw\rangle=$
$-v_{0}dw_{0}+ \sum_{i=1}^{n+1}v_{i}dw_{i}$
on
$\mathbb{R}_{1}^{n+2}\cross \mathbb{R}_{1}^{n+2}$ and consider the following two double fibrations:(1)(a) $LC^{*}\cross S_{1}^{n+1}\supset\Delta_{3}=\{(v, w)|\langle v, w\rangle=1\},$ (b) $\pi_{31}:\triangle_{3}arrow LC^{*},\pi_{32}:\Delta_{3}arrow S_{1}^{n+1},$
(c) $\theta_{31}=\langle dv,$$w\rangle|\Delta_{3},$$\theta_{32}=\langle v,$$dw\rangle|\Delta_{3}.$
(2) (a) $LC^{*}\cross LC^{*}\supset\Delta_{4}=\{(v, w)|\langle v, w\rangle=-2\},$ (b) $\pi_{41}:\Delta_{4}arrow LC^{*},\pi_{42}:\Delta_{4}arrow LC^{*},$ (c) $\theta_{41}=\langle dv,$$w\rangle|\Delta_{4},$$\theta_{42}=\langle v,$$dw\rangle|\Delta_{4}.$
Here, $\pi_{i1}(v, w)=v,$ $\pi_{i2}(v, w)=w$
.
We remark that $\theta_{i1}^{-1}(0)$ and $\theta_{i2}^{-1}(0)$ define the samethat each $(\Delta_{i}, K_{i})(i=3,4)$ is a contact manifold and both of$\pi_{ij}(j=1,2)$ areLegendrian
fibrations. Moreover thosecontact manifolds are contact diffeomorphic to each other. In [3]
we have defined four double fibrations $(\triangle_{i}, K_{i})(i=1,2,3,4)$ such that these are contact
diffeomorphic to each other. Here, we only use $(\triangle_{3}, K_{3})$ and $(\triangle_{4}, K_{4})$
.
If we have anisotropic mapping $i$ : $Larrow\Delta_{i}$ $(i.e., i^{*}\theta_{i1}=0)$, we say that $\pi_{i1}(i(L))$ and $\pi_{i2}(i(L))$ are
$\Delta_{i}$-dual to each other $(i=3,4)$
.
For detailed properties ofLegendrian fibrations, see [1].Wenow define hypersurfaces in $LC^{*}$ associated with the hypersurfaces in $S_{+}^{n}$ or $S_{0}^{n}$
.
Let$x$ : $Uarrow S_{+}^{n}$ be ahypersurface. We define$\overline{LD}\frac{\pm}{M}$ : $U\cross \mathbb{R}arrow LC^{*}$ by
$\overline{LD}\frac{\pm}{M}(u,\mu)=\overline{x}(u)+\mu n(u)\pm\sqrt{\mu^{2}+1}e_{0}.$
We also define $LD_{M}$ : $U\cross \mathbb{R}arrow LC^{*}$ by
$LD_{M}(u, \mu)=(\mu^{2}/4-1)\overline{x}(u)+\mu n(u)+(\mu^{2}/4+1)e_{0}.$
Then we have the following proposition.
Proposition 3.1. Under the above notation, we have the followings: (1) $\overline{x}$ and$\overline{LD}\frac{\pm}{M}$ are $\Delta_{3}$-dualto each other.
(2) $x$ and$LD_{M}$ are$\Delta_{4}$-dual to each other.
We call each one of$\overline{LD}\frac{\pm}{M}$ the lightcone dual hypersurface along
$\overline{M}\subset S_{0}^{n}$ and $LD_{M}$ the
lightconedual hypersurface along$M\subset S_{+}^{n}$
.
Thenwehavetwo mappings$\pi 0\overline{LD}\frac{\pm}{M}$
: $U\cross \mathbb{R}arrow$
$S_{+}^{n}$ and $\pi oLD_{M}$ : $U\cross \mathbb{R}arrow S_{+}^{n}$ defined by
$\pi\circ\overline{LD}\frac{\pm}{M}(u, \mu) = \pm(\frac{1}{\sqrt{\mu^{2}+1}}\overline{x}(u)+\frac{\mu}{\sqrt{\mu^{2}+1}}n(u))+e_{0},$
$\pi\circ LD_{M}(u, \mu) = \frac{\mu^{2}-4}{\mu^{2}+4}\overline{x}(u)+\frac{4\mu}{\mu^{2}+4}n(u)+e_{0}.$
Let $x$ : $Uarrow S_{+}^{n}$ be a hypersurface in the lightlike unit sphere. Then we define two
families offunctions as follows:
$\overline{H}$ : $U\cross LC^{*}arrow \mathbb{R}$;
$\overline{H}(u,\overline{v})=\langle\overline{x}(u),\overline{v}\rangle-1,$
$H:U\cross LC^{*}arrow \mathbb{R}$; $H(u, v)=\langle x(u),$$v\rangle+2.$
We$cal1\overline{H}$alightcone height
function
of the deSitterspherical hypersurface M. For anyfixed$\overline{v}_{0}\in LC^{*}$, wedenote $\overline{h}_{\overline{v}_{0}}(u)=\overline{H}(u, \overline{v}_{0})$. We also call $H$ a $light\omega ne$ height
function
ofthe lightlike spherical hypersurface $M$.
For any fixed $v_{0}\in LC^{*}$, wedenote $h_{v_{0}}(u)=H(u, v_{0})$.
Proposition 3.2. Let $\overline{M}$ be a hypersurface in$S_{0}^{n}$ and$\overline{H}$ the lightcone height
funct\’ion
onM.
For$p=x(u)$ and$\overline{p}=\overline{x}(u)\neq\overline{v}^{\pm}$, we have the followmgs:(1) $\overline{h}_{\overline{v}}\pm(u)=\partial\overline{f}_{b_{v}}\pm/\partial u_{i}(u)=0(i=1, \ldots,n-1)$\’ifand only
if
$\overline{v}^{\pm}=\overline{LD}\frac{\pm}{M}(u, \mu)$
for
some $\mu\in \mathbb{R}\backslash \{0\}.$(2} $\overline{h}_{\overline{v}}\pm(u)=\partial\overline{h}_{\overline{v}}\pm/\partial u_{i}(u)=0(i=1, \ldots, n-1)$ and$\det$ Hess $(\overline{h}_{\overline{v}}\pm)(u)=0$
if
and onlyif
$\overline{v}^{\pm}=\overline{LD}\frac{\pm}{M}(u, \mu),$ $1/\mu$ is oneProposition 3.3. Let$M$ be a hypersurface in$S_{+}^{n}$ and $H$bethe lightcone height
function
onM. For$p=x(u)\neq v$, we have the follounngs.
(1) $h_{v}(u)=\partial h_{v}/\partial u_{i}(u)=0,$ $(i=1, \ldots,n-1)$
if
andonlyif
$v=LD_{M}(u, \mu)$
for
some $\mu\in \mathbb{R}\backslash \{0\}.$(2) $h_{v}(u)=\partial h_{v}/\partial u_{i}(u)=0,$ $(i=1, \ldots,n-1)$ and $\det$ Hess $(h_{v})(u)=0$
if
and onlyif
$v=LD_{M}(u, \mu),$ $(\mu/4-1/\mu)w$ one the non-zero principle curvatures $\kappa_{i}(p)$
of
$M.$Let $(u, \mu)$ be
a
singularpoint of eachone
of$\overline{LD}\frac{\pm}{M}$.
By Proposition 3.2,
we
have $1/\mu=$$\kappa_{i}(p)(1\leq i\leq n-1)$, where $\kappa_{i}(p)$ is one of the
non-zero
principle curvatures of $M$ at$p=x(u)$. It follows that $\mu=1/\kappa_{i}(p)$
.
Therefore the critical value sets of$\overline{LD}\frac{\pm}{M}$ are givenby
$C( \overline{LD}\frac{\pm}{M})=\bigcup_{i=1}^{n-1}\{\overline{x}(u)+\frac{1}{\kappa_{i}(p)}n(u)\pm\sqrt{\frac{1}{\kappa_{i}^{2}(p)}+1}e_{0}|u\in U\}.$
Let $(u, \mu)$ be a singular point of$LD_{M}(u, \mu)$
.
By Proposition 3.3, we have $\mu/4-1/\mu=$$\kappa_{i}(p)(1\leq i\leq n-1)$. It follows that wehave$\mu=2(\kappa_{i}(p)\pm\sqrt{1+\kappa_{i}^{2}(p)})$
.
For simplification,we write that $\sigma^{\pm}(\kappa_{i}(p))=\kappa_{i}(p)\pm\sqrt{1+\kappa_{i}^{2}(p)}$. Then the critical value sets of $LD_{M}$ are given by
$C(LD_{M})^{\pm}= \bigcup_{i=1}^{n-1}\{((\sigma^{\pm}(\kappa_{i}(p)))^{2}-1)\overline{x}(u)+2\sigma^{\pm}(\kappa_{i}(p))n(u)+((\sigma^{\pm}(\kappa_{i}(p)))^{2}+1)e_{0}|u\in U\}.$
We respectively denote that
$LF \frac{\pm}{M}=\bigcup_{i=1}^{n-1}\{\overline{x}(u)+\frac{1}{\kappa_{i}(p)}n(u)\pm\sqrt{\frac{1}{\kappa_{i}^{2}(p)}+1}e_{0}|u\in U\},$
$LF_{M}^{\pm}= \bigcup_{i=1}^{n-1}\{((\sigma^{\pm}(\kappa_{i}(p)))^{2}-1)\overline{x}(u)+2\sigma^{\pm}(\kappa_{i}(p))n(u)+((\sigma^{\pm}(\kappa_{i}(p)))^{2}+1)e_{0}|u\in U\}.$
We respectively call each one of $LF^{\underline{\pm}}$ the lightcone
focal surface
ofthe de Sitter sphericalhypersurface $\overline{x}$ and each one of $LF_{M}\ovalbox{\tt\small REJECT}$
the ligtcone
focal surface
of the lightcone spherical hypersurface$x$. Then the projections of these surfaces to $S_{+}^{n}$are
givenas
follows:$\pi(C(\overline{LD}\frac{\pm}{M}))=\bigcup_{i=1}^{n-1}\{\pm(\sqrt{\frac{\kappa_{i}^{2}(p)}{1+\kappa_{i}^{2}(p)}}\overline{x}(u)+\sqrt{\frac{1}{1+\kappa_{i}^{2}(p)}}n(u))+e_{0}|u\in U\},$
$\pi(C(LD_{M})^{\pm})=\bigcup_{i=1}^{n-1}\{\frac{(\sigma^{\pm}(\kappa_{i}(p)))^{2}-1}{(\sigma^{\pm}(\kappa_{i}(p)))^{2}+1}\overline{x}(u)+\frac{2\sigma^{\pm}(\kappa_{i}(p))}{(\sigma^{\pm}(\kappa_{i}(p)))^{2}+1}n(u)+e_{0}|u\in U\}.$
Bydefinition, wehave$\epsilon\frac{\pm}{M}=\Phi\circ\pi(C(\overline{LD}\frac{\pm}{M}))$, whereeach oneof
$\epsilon\frac{\pm}{M}$is thespherical evolute
sets of thelightconedual hypersurfacesof$\overline{M}=\overline{x}(U)$.Since$\sigma^{\pm}(\kappa_{i}(p))=\kappa_{i}(p)\pm\sqrt{1+\kappa_{i}^{2}(p)},$ wehave$(\sigma^{\pm}(\kappa_{i}(p)))^{2}=2\kappa_{i}(p)\sigma^{\pm}(\kappa_{i}(p))+1$
.
By straightforward calculations, we have$( \frac{(\sigma^{\pm}(\kappa_{i}(p)))^{2}-1}{(\sigma^{\pm}(\kappa_{i}(p)))^{2}+1})^{2}=\frac{\kappa_{i}^{2}(p)(\sigma^{\pm}(\kappa_{i}(p)))^{2}}{\kappa_{i}^{2}(p)(\sigma^{\pm}(\kappa_{i}(p)))^{2}+(\sigma^{\pm}(\kappa_{i}(p)))^{2}}=\frac{\kappa_{i}^{2}(p)}{1+\kappa_{i}^{2}(p)}$
and
$( \frac{2\sigma^{\pm}(\kappa_{i}(p))}{(\sigma^{\pm}(\kappa_{i}(p)))^{2}+1})^{2}=\frac{(\sigma^{\pm}(\kappa_{i}(p)))^{2}}{\kappa_{i}^{2}(p)(\sigma^{\pm}(\kappa_{i}(p)))^{2}+(\sigma^{\pm}(\kappa_{i}(p)))^{2}}=\frac{1}{1+\kappa_{i}^{2}(p)}.$
Thus we have the following proposition.
Proposition 3.4. Let$x$ : $Uarrow S_{+}^{n}$ be ahypersurface in$S_{+}^{n}$
.
ThenWe define$\tilde{\pi}=\Phi 0\pi$ : $LC^{*}arrow S_{0}^{n}$
.
Then we havethe following theoremas
acorollary of Proposition 3.4.Theorem 3.5. Both
of
the projectionsof
the critical volue sets $C( \overline{LD}\frac{\pm}{M})$ and $C(LD_{M})^{\pm}$in the unit sphere $S_{0}^{n}$ are the images
of
the sphericalevolutesof
$M.$$\tilde{\pi}(C(\overline{LD}\frac{\pm}{M}))=\tilde{\pi}(C(LD_{M})^{\pm})=\epsilon\frac{\pm}{M}.$
4.
The lightcone dual hypersurfaces
as
wave
fronts
Wenow naturally interpret the lightcone dual hypersurfaces of the submanifolds in $S_{+}^{n}$ as
wavefrontsets in the theory of Legendrian singularities. Let $\overline{\pi}$:$PT^{*}(LC^{*})arrow LC^{*}$ be the
projective cotangentbundleswith canonicalcontactstructures. Considerthetangent bundle
$\tau$ : $TPT^{*}(LC^{*})arrow PT^{*}(LC^{*})$ and the differential map $d\overline{\pi}:TPT^{*}(LC^{*})arrow T(LC^{*})$ of
$\overline{\pi}$
.
For any$X\in TPT^{*}(LC^{*})$, there exists an element $\alpha\in T^{*}(LC^{*})$ such that $\tau(X)=[\alpha].$
For an element $V\in 1_{v}(LC^{*})$, the property $\alpha(V)=0$ dose not depend on the choice of
representative of the class $[\alpha]$
.
Thus we have the canonical contact structure on$PT^{*}(LC^{*})$by
$K=\{X\in TPT^{*}(LC^{*})|\tau(X)(c\Gamma\pi(X))\}=0.$
coordinate neighborhood $(U, (\pm\sqrt{v_{1}^{2}++v_{n+1}^{2}}, v_{1}, \ldots,v_{n+1}))$ in $LC^{*}$, we have a
trivial-ization $PT^{*}(LC^{*})\equiv LC^{*}\cross P(\mathbb{R}^{n})^{*}$ and we call $((\pm\sqrt{v_{1}^{2}++v_{n+1}^{2}}, v_{1}, \ldots, v_{n+1}),$ $[\xi_{1}$ :
. .
.
: $\xi_{n+1}])$ homogeneous coordinates of$PT^{*}(LC^{*})$, where $[\xi_{1} :. . .: \xi_{n+1}]$ arethe homoge-neous coordinates of the dual projective space $P(\mathbb{R}^{n})^{*}$.
It is easy to show that $X\in K_{(v,[\xi])}$if and only if $\sum_{1}^{n+1}\mu_{i}\xi_{i}=0$, where $c f\overline{\pi}(X)=\sum_{1}^{n+1}\mu_{i}\partial/\partial v_{i}\in 1_{v}’LC^{*}$. An immersion $i:Larrow PT^{*}(LC^{*})$ issaid to be aLegendrianimmersionif$\dim L=n$and$\dot{d}i_{q}(T_{q}L)\subset K_{i(q)}$ for any $q\in L$
.
The map To$i$ is also called the Legendrian map and we call the set $W(i)=image\overline{\pi}\circ i$ thewave front of$i$. Moreover, $i$(lift of $W(i)$. Let $F$ : $(\mathbb{R}^{k}\cross \mathbb{R}^{n}, 0)arrow(\mathbb{R}, 0)$ be a function germ. We say that $F$ is a Morse family of hypersurfaces if the map germ $\Delta^{*}F$ : $(\mathbb{R}^{k}\cross \mathbb{R}^{n}, 0)arrow(\mathbb{R}^{k+1},0)$ defined by $\Delta^{*}F’=(F, \partial F/\partial u_{1}, \cdots, \partial F/\partial u_{k})$
.
is nonsingular. In this case, we have the followingsmooth $(n-1)$-dimensional smoothsubmanifold.
$\Sigma_{*}(F)=\{(u, v)\in(\mathbb{R}^{k}\cross \mathbb{R}^{n}, 0)|F(u, v)=\frac{\partial F}{\partial u_{1}}(u, v)=\cdots=\frac{\partial F}{\partial\eta 1_{k}}(u, v)=0\}=(\Delta^{*}F)^{-1}(0)$
.
The map germ $\mathcal{L}_{F’}$ : $(\Sigma_{*}(F’), 0)arrow P’1’*\mathbb{R}^{n}$ defined by
$\mathcal{L}_{F’}(u, v)=(v, [\frac{\partial F}{\partial v_{1}}(u, v)$ :.. . : $\frac{\partial F}{\partial v_{n}}(u, v)])$
.
is a Legendrian immersion germ. Then we have the following fundamental theorem of
Amol’d and Zakalyukin [1, 14].
Proposition4.1. AllLegendrian
submanifold
germsin$PT^{*}\mathbb{R}^{n}$ areconstructed by the abovemethod.
We call $F$ a generatingfamily of$\mathcal{L}_{F}(\Sigma_{*}(F))$
.
Therefore the wave front of$\mathcal{L}_{F}$ is$W(\mathcal{L}_{F})=\{v\in \mathbb{R}^{n}|\exists u\in \mathbb{R}^{k}$such that $F(u, v)= \frac{\partial F}{\partial u_{1}}(u, v)=\ldots=\frac{\partial F}{\partial u_{k}}(u, v)=0\}.$ We claim here that we have atrivialization
as
follows:$\Phi$ : $PT^{*}(LC^{*}) \equiv LC^{*}\cross P(\mathbb{R}^{n})^{*};\Phi([\sum_{i=1}^{n+1}\xi_{i}dv_{i}])=(v_{0}, v_{1}, \cdots, v_{n+1}),$ $[\xi_{1} :. .. \xi_{n+1}])$ by using the above coordinate system.
Proposition 4.2. The lightcone height
function
$H$ : $U\cross LC^{*}arrow \mathbb{R}$ is a Morse famdyof
the hypersurface around $(u, v)\in\Sigma_{*}(H)$
.
We also have the following proposition.
Proposition 4.3. The lightcone height
function
$\overline{H}$ : $U\cross LC^{*}arrow \mathbb{R}$ is aMorse $fam\iota ly$
of
the hypersurface around $(u, v)\in\Sigma_{*}(\overline{H})$. Here,
we
consider the Legendrian immersion$\mathcal{L}_{4}$ : $(u, \mu)arrow\triangle_{4};\mathcal{L}_{4}(u, \mu)=(LD_{M}(u, \mu),x(u))$
.
We define the following mapping:
$\Psi$ :
$\Delta_{4}arrow LC^{*}\cross P(\mathbb{R}^{n})^{*};\Psi(v, w)=(v, [v_{0}w_{1}-v_{1}w_{0} :. . . :v_{0}w_{n+1}-v_{n+1}w_{0}])$
.
For the canonical contact form $\theta=\sum_{i=1}^{n+1}\xi_{i}dv_{i}$ on $PT^{*}(LC^{*})$, we have $\Psi^{*}\theta=(v_{0}w_{1}-$ $v_{1}w_{0})dv_{1}+\cdots+(v_{0}w_{n+1}-v_{n+1}w_{0})dv_{n+1}|_{\Delta_{4}}=v_{0}(-w_{0}dv_{0}+w_{1}dv_{1}+\cdots+w_{n+1}dv_{n+1})-$ $w_{0}(-v_{0}dv_{0}+v_{1}dv_{1}+\cdots+v_{n+1}dv_{n+1})|_{\Delta_{4}}=v_{0}\langle w,$$dv\rangle|_{\Delta_{4}}=v_{0}\theta_{42}|_{\triangle_{4}}$. Thus $\Psi$ is a contact
morphism.
Theorem 4.4. For any hypersurface $x$ : $Uarrow S_{+}^{n}$, the lightcone height
function
$H$ : $U\cross LC^{*}arrow \mathbb{R}$ is agenerating familyof
the Legendnan immersion $\mathcal{L}_{4}.$Similarly,weconsider theLegendrianimmersions$\mathcal{L}_{3}^{\pm}:(u,\mu)arrow\Delta_{3}$ definedby$\mathcal{L}_{3}^{\pm}(u,\mu)=$ $( \overline{LD}\frac{\pm}{M}(u, \mu),\overline{x}(u))$
.
Then wehave the following theorem.Theorem 4.5. For any hypersurface $\overline{x}$ : $Uarrow S_{0}^{n}$, the lightcone height
function
$\overline{H}$ : $U\cross LC^{*}arrow \mathbb{R}$ is agenerating familyof
the Legendrian immersions $\mathcal{L}_{3}^{\pm}.$5.
Contact
with parabolic
$(n-1)$
-spheres
and parabolic
$n$
-hyperquadrics
Beforewe startto consider thecontact between hypersurfaces in the sphere with parabolic
$(n-1)$-sphere and parabolic $n$-hyperquadrics, we briefly reviewthe theory ofcontact due
to Montaldi[8]. Let $X_{i},$$Y_{i}(i=1,2)$ be submanifolds of $\mathbb{R}^{n}$ with
$\dim X_{1}=\dim X_{2}$ and
$\dim Y_{1}=\dim Y_{2}$
.
We say that the contact of$X_{1}$ and$Y_{1}$ at $y_{1}$ is thesame typeas the $\omega$ntactof
$X_{2}$ and$Y_{2}$ at$y_{2}$ if thereisa diffeomorphism$\Phi$ : $(\mathbb{R}^{n}, y_{1})arrow(\mathbb{R}^{n}, y_{2})$ such that $\Phi(X_{1})=X_{2}$ and $\Phi(Y_{1})=Y_{2}$. In this case, we write $K(X_{1}, Y_{1};y_{1})=K(X_{2}, Y_{2};y_{2})$
.
Of course, in thedefinition, $\mathbb{R}^{n}$ canbereplaced by any manifold. Two function germs
$f_{i}$ : $(\mathbb{R}^{n}, a_{i})arrow \mathbb{R}(i=$
$1,2)$ arecalled $\mathcal{K}$-equivalent if there is a diffeomorphismgerm $\Phi$ : $(\mathbb{R}^{n}, a_{1})arrow(\mathbb{R}^{n}, a_{2})$, and
a function germ $\lambda$ : $(\mathbb{R}^{n}, a_{1})arrow \mathbb{R}$ with $\lambda(a_{1})\neq 0$ such that
$f_{1}=\lambda\cdot(f_{2}o\Phi)$
.
Theorem5.1 (Montaldi[8]). Let$X_{i},$ $Y_{i}$$(for i=1,2)$ be
submanifol& of
$\mathbb{R}^{n}$ with$dimX_{1}=dimX_{2}$and $dimY_{1}=dimY_{2}$. Let $g_{i}$ : $(X_{i}, x_{i})arrow(\mathbb{R}^{n}, y_{i})$ be immersiongerms and$f_{i}$ : $(\mathbb{R}^{n}, y_{i})arrow$
$(\mathbb{R}^{p}, 0)$ besubmersiongermswith$(Y_{i}, y_{i})=(f_{i}^{-1}(0), y_{i})$. Then$K(X_{1}, Y_{1};y_{1})=K(X_{2}, Y_{2};y_{2})$
if
and $07dy$if
$f_{1}og_{1}$ and$f_{2}og_{2}$ are $\mathcal{K}$-equivalent.Retuming to the lightcone dual hypersurface $LD_{M}$, we
now
consider the function $\mathfrak{h}$ :$S_{+}^{n}\cross LC^{*}arrow \mathbb{R}$ defined by $\mathfrak{h}(u, v)=\langle u,$$v\rangle+2$ and the function $\mathfrak{g}$ : $LC^{*}\cross LC^{*}arrow \mathbb{R}$
defined by $\mathfrak{g}(u, v)=\langle u,$ $v\rangle+2$ . For a given $v_{0}\in LC^{*}$, we denote $\mathfrak{h}_{v_{0}}(u)=\mathfrak{h}(u, v_{O})$ and $\mathfrak{g}_{v_{U}}(u)=\mathfrak{g}(u, v_{0})$, then we have$\mathfrak{h}_{v_{0}}^{-1}(0)=S_{+}^{n}\cap HP(v_{O}, -2)$ and$\mathfrak{g}_{v_{0}}-1(0)=LC^{*}\cap HP(v_{O}, -2)$
.
Forany $u_{0}\in U,$ $\mu_{0}\in \mathbb{R}$, wetake thepoint $v_{0}=LD_{M}(u_{0}, \mu_{0})$. Then we have
$\mathfrak{g}_{v_{0}}ox(u_{0})=\mathfrak{g}o(x\cross id_{LC^{*}})(u_{0}, v_{0})=\mathfrak{h}_{v_{0}}ox(u_{0})=\mathfrak{h}\circ(x\cross id_{LC^{*}})(u_{0}, v_{0})=H(u_{0}, v_{0})=0.$
We also have
$\frac{\partial(\mathfrak{g}_{v_{0}}\circ x)}{\partial u_{i}}(u_{0})=\frac{\partial(\mathfrak{h}_{v_{0}}\circ x)}{\partial u_{i}}(u_{0})=\frac{\partial H}{\partial u_{i}}(u_{0}, v_{0})=0$
for$i=1,$ $\cdots,$$n-1$
.
This meansthat the $(n-1)$-sphere$\mathfrak{h}_{v_{0}}^{-1}(0)=S_{+}^{n}\cap HP(v_{0}, -2)$ istangentto $M=x(U)$ at$p_{0}=x(u_{0})$. Inthis case, we call it the lightcone tangent parabolic $(n-1)-$
sphere of $M$ at $p_{0}$, which is denoted by $TPS_{+}^{n-1}(x, u_{0})$. The $n$-hyperquadric $\mathfrak{g}_{v_{0}}^{-1}(0)=$
$LC^{*}\cap HP(v_{O}, -2)$ is alsotangent to $M$ at $p_{0}$
.
In thiscase, wecall it the lightcone tangentparabolic$n$-hyperquadric of $M$ at $p_{0}$, which is denoted by $TPH^{}$ $(x, u_{0})$. Forthe lightcone
dual surfaces $\overline{LD}\frac{\pm}{M}$, we consider a function $\overline{\mathfrak{h}}$ :
$S_{0}^{n}\cross LC^{*}arrow \mathbb{R}$ defined by $\overline{\mathfrak{h}}(u, v)=$ $\langle u,$$v\rangle-1$ and a function $\overline{\mathfrak{g}}.$ $S_{1}^{n+1}\cross LC^{*}arrow \mathbb{R}$ defined by $\overline{\mathfrak{g}}(u, v)=\langle u,$$v\rangle-1$ For a
given $v_{0}\in LC^{*}$, we denote that $\overline{\mathfrak{h}}_{v_{0}}(u)=\overline{\mathfrak{h}}(u, v_{O})$ and $\overline{\mathfrak{g}}_{v0}(u)=\overline{\mathfrak{g}}(u, v_{0})$
.
Then we have$\overline{\mathfrak{h}_{v0}}^{1}(0)=S_{0}^{n}\cap HP(v_{O}, 1)$ and $\overline{\mathfrak{g}_{v_{0}}}^{1}(0)=S_{1}^{n+1}\cap HP(v_{0},1)$
.
For any $u_{0}\in U$ and the points$\overline{v}_{0}^{\pm}=\overline{LD}\frac{\pm}{M}(u_{0}, \mu_{0})$, wehave
$\overline{\mathfrak{g}}_{\overline{v}_{\cup}}\pm\circ\overline{x}(u_{0})=\overline{\mathfrak{g}}\circ(\overline{x}\cross id_{LC^{*}})(u_{0}, \overline{v}_{0}^{\pm})=\overline{\mathfrak{h}}_{\overline{v}_{\cup}}\pm\circ\overline{x}(u_{0})=\overline{\mathfrak{h}}\circ(\overline{x}\cross id_{LC^{*}})(u_{0}, \overline{v}_{0}^{\pm})=\overline{H}(u_{0}, \overline{v}_{0}^{\pm})=0.$
We also have
$\frac{\partial(\overline{\mathfrak{g}}_{\overline{v}_{0}^{\pm}}\circ\overline{x})}{\partial u_{i}}(u_{0})=\frac{\partial(\overline{\mathfrak{h}}_{\overline{v}_{0}^{\pm}}\circ\overline{x})}{\partial v_{i}}(u_{0})=\frac{\partial\overline{H}}{\partial u_{i}}(u_{0}, \overline{v}_{0}^{\pm})=0$
for$i=1,$$\cdots,$$n-1$. It follows that eachoneof the $(n-1)-$sphere$\overline{\mathfrak{h}_{\overline{v}_{0}}}^{1}\pm(0)=S_{0}^{n}\cap HP(\overline{v}_{0}^{\pm}, 1)$
is tangent to $\overline{M}$ at
$\overline{p}_{0}=\overline{x}(u_{0})$
.
In this case, we call each one the tangent parabolic$(n-1)$-sphere of$\overline{M}$
the n-hyperquadric$\overline{\mathfrak{g}}_{\overline{v}_{0}^{\pm}}^{-1}(0)=S_{1}^{n+1}\cap HP(\overline{v}_{0}^{\pm}, 1)$ is tangent to $\overline{M}$at
$\overline{p}_{0}$
.
In this case,we
calleach onethe de-Sitter tangent parabolic $n$-hyperquadric of$\overline{M}$at
$\overline{p}_{0}$, which are denoted by
$TPS_{1}^{n\pm}(\overline{x},u_{0})$.
Let $x_{i}$ : $(U, u_{i})arrow(S_{+}^{n},p_{i})(i=1,2)$ be hypersurface germs. For $v_{l}\prime=LD_{M_{t}},$$(u_{i}, \mu_{i})$,
we denote $h_{i,v_{i}}$ : $(U, u_{i})arrow(\mathbb{R}, 0)$ by $h_{i,v_{i}}(u_{i})=H(u_{i}, v_{i})$
.
Thenwe
have $h_{i,v_{t}}(u)=$$(\mathfrak{h}_{i,v_{i}}ox_{i})(u)=(\mathfrak{g}_{i,v_{i}}ox_{i})(u)$
.
For$\overline{v}_{i}^{\pm}=\overline{LD}\frac{\pm}{M}i(u_{i}, \mu_{i})$, We denote$\overline{h}_{i,\overline{v}_{t}}\pm:(U, u_{i})arrow(\mathbb{R}, 0)$
by $\overline{h}_{i,\overline{v}_{i}^{\pm}}(u_{i})=\overline{H}(u_{i},\overline{v}_{i}^{\pm})$
.
Then wehave $\overline{h}_{i,\overline{v}_{i}}\pm(u)=(\overline{\mathfrak{h}}_{i,\overline{v}_{i}}\pm\circ\overline{x}_{i})(u)=(\overline{\mathfrak{g}}_{i,\overline{v}_{i}}\pm\circ\overline{x}_{i})(u)$.
ByTheorem 5.1, we have the following proposition.
Proposition 5.2. Let $x_{i}$ : $(U, u_{i})arrow(S_{+}^{n},p_{i})(i=1,2)$ be hypersurface germs. For $v_{i}=$
$LD_{M_{:}}(u_{i},\mu_{i})$, the follovring conditions are equivalent:
(1) $K(x_{1}(U),TPS_{+}^{n-1}(x_{1}, u_{1}), v_{1})=K(x_{2}(U),TPS_{+}^{n-1}(x_{2}, u_{2}), v_{2})$
.
(2) $K(x_{1}(U),TPH^{n}(x_{1}, u_{1}), v_{1})=K(x_{2}(U),$TPH $(x_{2}, u_{2}),$$v_{2})$.(3) $h_{1,v_{1}}$ and$h_{2,v_{2}}$
are
$\mathcal{K}$-equivalent.Moreover,
for
$\overline{v}_{it\prime}^{\pm_{=\overline{LD}\frac{\pm}{M}}}(u_{i},\mu_{i})$, thefollounng$\omega$nditions
are
equivalent:(4) $K(x_{1}(U)_{)}TPS_{0}^{n-1\pm}(x_{1}, u_{1}), \overline{v}_{1}^{\pm})=K(x_{2}(U), TPS_{0}^{n-1\pm}(x_{2}, u_{2})_{)}\overline{v}_{2}^{\pm})$.
(5) $K(x_{1}(U),TPS_{1}^{n\pm}(x_{1}, u_{1}),\overline{v}_{1}^{\pm})=K(x_{2}(U)_{)}TPS_{1}^{n\pm}(x_{2}, u_{2}), \overline{v}_{2}^{\pm})$ .
(6) $\overline{h}_{1,\overline{v}_{1}}\pm$ and$\overline{h}_{2,\overline{v}_{2}}\pm are\mathcal{K}$-equival$ent.$
On the otherhand, w\‘ereturnto the review
on
thetheory of Legendrian singularities. Weintroduce a natural equivalence relation among Legendrian submanifold germs. Let $F,$ $G$ :
$(\mathbb{R}^{k}\cross \mathbb{R}^{n}, 0)arrow(\mathbb{R}, 0)$ be Morse families of hypersurfaces. Then we say that $\mathcal{L}_{F}(\Sigma_{*}(F))$
and $\mathcal{L}_{G}(\Sigma_{*}(G))$ are Legendrian equivalent if there exists a contact diffeomorphism germ
$H$ : $(PT^{*}\mathbb{R}^{n}, z)arrow(PT^{*}\mathbb{R}^{n}, z’)$ such that$H$preserves fibersof$\pi$and that$H(\mathcal{L}_{F’}(\Sigma_{*}(F’)))=$
$\mathcal{L}_{G}(\Sigma_{*}(G))$, where $z=\mathcal{L}_{F’}(0),$$z’=\mathcal{L}_{G}(0)$
.
By using the Legendrian equivalence, wecan
definethe notion ofLegendrian stability for Legendrian submanifold germsby the ordinary way (see, [l][Part III]). We
can
interpret the Legendrian equivalence by usingthe notion of generating families. We denote by $\mathcal{E}_{n}$ the local ring offunction germs $(\mathbb{R}^{n}, 0)arrow \mathbb{R}$ withthe unique maximal ideal $\mathfrak{M}_{n}=\{h\in \mathcal{E}_{n}|h(O)=0\}$
.
Let $F,$$G:(\mathbb{R}^{k}\cross \mathbb{R}^{n},0)arrow(\mathbb{R}, 0)$be function germs.Let $Q_{n+1}(x, u_{0})$ be thelocal ring of the function germ $h_{v_{0}}$ : $(U, u_{0})arrow \mathbb{R}$ defined by
$Q_{n+1}(x,u_{0})=C_{u_{U}}^{\infty}(U)/(\langle h_{v_{0}}\rangle_{C_{u}^{\infty}(U)}+\mathfrak{M}_{n-1}^{n+2})0,$
and $Q_{n+1}^{\pm}(\overline{x}, u_{0})$be the local rings of the function germs$\overline{h}_{\overline{v}_{0}^{\pm}}$ : $(U,u_{0})arrow \mathbb{R}$ defined by $Q_{n+1}^{\pm}(\overline{x}, u_{0})=C_{u_{0}}^{\infty}(U)/(\langle\overline{h}_{\overline{v}_{U}^{\pm}}\rangle_{C_{u_{0}}^{\infty}(U)}+\mathfrak{M}_{n-1}^{n+2})$,
where$v_{0}=LD_{M}(u_{0}, \mu_{0}),$$\overline{v}_{0}^{\pm}=\overline{LD}\frac{\pm}{M}(u_{0}, \mu_{0})$and
$C_{u0}^{\infty}(U)$is the localringoffunction germs
at $u_{0}$ with theuniquemaximal ideal $\mathfrak{M}_{n-1}.$
Theorem 5.3. Let $x_{i}$ : $(U, u_{i})arrow(S_{+}^{n},p_{i})(i=1,2)$ be hypersurface germs such that
the $\omega$rresponding Legendrian immersion germs are Legendrian stable. Then the following
conditions are equivalent.
(1) The lightcone hypersurface germs $LD_{M_{1}}(U\cross \mathbb{R})$ and$LD_{M_{2}}(U\cross \mathbb{R})$ arediffeomorphic. (2) Legendrian immersiongerms$\mathcal{L}_{4}^{1}$ and $\mathcal{L}_{4}^{2}$ are Legendrian equivalent.
(3) The lightcone height
functions
germs $H_{1}$ and$H_{2}$ are$\mathcal{P}-\mathcal{K}$-equivalent.(4) $h_{1,v_{1}}$ and$h_{2,v}2$ are $\mathcal{K}$-equivalent.
(6) $K(x_{1}(U),$TPH $(x_{1}, u_{1}),$$v_{1})=K(x_{2}(U),$TPH $(x_{2}, u_{2}),$$v_{2})$.
(7) Local rings$Q_{n+1}(x_{1}, u_{1})$ and$Q_{n+1}(x_{2}, u_{2})$ are isomorphic as$\mathbb{R}$-algebras.
Theorem 5.4. Let $\overline{x}_{i}$ : $(U, u_{i})arrow(S_{0}^{n},p_{i})(i=1,2)$ be hypersurface germs such that
the $\omega$rresponding Legendrian immersion germs are Legendnan stable. Then the following
conditions are equivalent.
(1) The lightcone hypersurface germs$\overline{LD}\frac{\pm}{M}1(U\cross \mathbb{R})and\overline{LD}\frac{\pm}{M}2(U\cross \mathbb{R})$ are diffeomorphic.
(2) Legendrian immersion germs $\mathcal{L}_{3}^{1\pm}$ and $\mathcal{L}_{3}^{2\pm}$ are Legendrian equivalent.
(3) The lightcone height
functions
germs $\overline{H}_{1}$ and$\overline{H}_{2}$ are$\mathcal{P}-\mathcal{K}$-equivalent.(4) $\overline{h}_{1,\overline{v}_{1}}\pm$ and$\overline{h}_{2,\overline{v}_{2}}\pm$ are
$\mathcal{K}$-equivalent.
(5) $K(\overline{x}_{1}(U), TPS_{0}^{n-1\pm}(\overline{x}_{1}, u_{1}), \overline{v}_{1}^{\pm})=K(\overline{x}_{2}(U), TPS_{0}^{n-1\pm}(\overline{x}_{2}, u_{2}), \overline{v}_{2}^{\pm})$
.
(6) $K(\overline{x}_{1}(U), TPS_{1}^{n\pm}(\overline{x}_{1}, u_{1}), \overline{v}_{1}^{\pm})=K(\overline{x}_{2}(U), TPS_{1}^{n\pm}(\overline{x}_{2}, u_{2}),\overline{v}_{2}^{\pm})$ .
(7) Localrings $Q_{n+1}^{\pm}(\overline{x}_{1}, u_{1})$ and$Q_{n+1}^{\pm}(\overline{x}_{2},u_{2})$ are isomorphic as$\mathbb{R}$-algebras.
Lemma 5.5. Let $x$ : $Uarrow S_{+}^{n}$ be a hypersurface germ such that the $\omega$rresponding
Leg-endrian immersion germs $\mathcal{L}_{4}$ and $\mathcal{L}_{3}^{\pm}$
are
Legendrian stable. Then at thesingular point
$v_{0}=LD_{M}(u_{0},2\sigma^{\pm}(\kappa_{i}(p_{0})))(1\leq i\leq n-1)$
of
$LD_{M}$ and the $sing_{lJ}lar$ points $\overline{v}_{0}^{\pm}=$$\overline{LD}\frac{\pm}{M}(u_{0},1/\kappa_{i}(p_{0}))of\overline{LD}\frac{\pm}{M}$, we have the following equivalent assertions:
(1) The lightcone hypersurface germs $LD_{M}(Ux\mathbb{R})$ and$\overline{LD}\frac{\pm}{M}(U\cross \mathbb{R})$ are diffeomorphic.
(2) Legendrian immersion germs $\mathcal{L}_{3}^{\pm}$ and$\mathcal{L}_{4}$ are Legendrian equivalent.
(3) Thelightcone height
functions
germs $H$ andi7
are $\mathcal{P}-\mathcal{K}$-equivalent.(4) $h_{v_{0}}$ and$\overline{h}_{\overline{v}_{\cup}}\pm$ are
$\mathcal{K}$-equivolent.
(5) $K(x(U), TPS_{+}^{n-1}(x, u_{0}), v_{0})=K(\overline{x}(U), TPS^{n-1\pm}(\overline{x}, u_{0}),\overline{v}_{0}^{\pm})$
.
(6) $K(x(U),\prime 1’PH^{n}(x, u_{0}), v_{0})=K(\overline{x}(U), TPS_{1}^{n}(\overline{x}, u_{0}),\overline{v}_{0}^{\pm})\ovalbox{\tt\small REJECT}.$
(7) Local rings $Q_{n+1}^{\pm}(\overline{x}, u_{0})$ and$Q_{n+1}(x, u_{0})$ are oeomorphic as $\mathbb{R}$-algebras.
By Lemma 5.5, we have our mainresult as the following theorem.
Theorem 5.6. Let $x_{i}$ : $(U, u_{i})arrow(S_{+}^{n},p_{i})(i=1,2)$ be hypersurface germs such that
the corresponding Legendrian immersion germs are Legendrian stable. At the singular points $\overline{v}_{i}^{\pm}=\overline{LD}\frac{\pm}{M}(u_{0},1/\kappa_{j}(p))(1\leqj\leq n-1)$
of
$\overline{LD}\frac{\pm}{M}$,and the $sing\tau rlar$ points $v_{i}=$ $LD_{M}(u_{0},2\sigma^{\pm}(\kappa_{j}(p)))$
of
$LD_{M}$, the $\omega$nditions ($1)\sim(7)$ in Theorem 5,3 and the conditions(1) $\sim(7)$ in Theorem
5.4
are all equivalent.References
[1] V. I.Amol’d,S. M. Gusein-Zade andA.N. Varchenko, Singvlanties
of
Differentiable
Mapsvol. I. Birkhauser, 1986.
[2] L. Chen and S. Izumiya, A mandala
of
Legendrian dualities forpseudo-spheres insemi-Euchdean space, Proceedings of the Japan Academy, 85 Ser. A, (2009), 49-54.
[3] S. Izumiya, Lengendrian dualities and spacelike hypersurfaces in the lightcone,Mosc.Math.
J. 9 (2009) 325-357.
[4] S. Izumiya,
Differential
Geometryfrom
the viewpointof
Lagrangian or Legendnansin-gularity theory, in Sinsin-gularity Theory (ed., D. Ch\’eniot et al), World Scientific (2007),
241-275.
[5] S. Izumiya, Y. Jiang and D.-H. Pei, Lightcone dualitiesforcurvesinthe sphere, The Quart.
[6] S. Izumiya, Y. Jiang andT. Sato, Lightcone dualities
for
curves in the 3-sphere, preprint (2012).[7] H. L. Liu, S. D. Jung, Hypersurfaces in lightlike cone, J. Geom. Phys. 58 (2008) 913-922.
[8] J.A.Montaldi,On ctact between subman\’ifolds, MichiganMath. J., 33(1986), 81-85.
[9] T. Nagai, The Gauss map
of
a hypersurface in Euclidean sphere and the sphericalLegen-drian duahty, Topologyandits Applications, 159 (2011) 545-554.
[10] T. Nagai, The sphencal evolute
of
ahypersurface inEuchdean sphere, in preparation.[11] I. R. Porteous, Some remrks on duality in $S^{3}$, Geometry and topology of caustics, Banach
Center Publ. 50, Polish Acad. Sci., Warsaw, (2004) 217-226.
[12] M. C. Romero Fuster, Sphere
stratifications
and the Gauss map, Proceedings of the RoyalSoc. Edinburgh, 95A (1983), 115-136.
[13] O. P. Shcherbak, Projectively dualspace curves and Legendre singularities, Sel. Math. Sov
5 (1986), 391-421.
[14] V. M. Zakalyukin, Lagrangian and Legendrian singularities, Funct. Anal. Appl. 10(1976),
No. 1, 23-31.
[15] S. Izumiya, Y. Jiang and D.-H. Pei, Lightcone dualities for hypersurface in the sphere,
preprint.
YANG JIANG
A. SCHOOL OF MATHEMATICS AND STATISTICS
NORTHEAST NORMAL UNIVERSITY
CHANGCHUN 130024 P. R. CHINA B. DEPARTMENT OF MATHEMATICS HOKKAIDO UNIVERSITY SAPPORO 060-OSIO JAPAN